First, I am an undergraduate so I apologize if this is trivial and certainly understand if it is closed immediately. 

I am currently in a combinatorics and graph theory class and recently we have been studying Hamiltonian graphs. We have been discussing a few theorems characterizing these graphs. I am interested in Dirac's theorem which states

> **Dirac (1952)** Let $G$ be a simple graph with $n \geq 3$ vertices such that for any vertex $v \in G$ we have $\deg(v) \geq \frac{n}{2}$. Then $G$ is Hamiltonian. 

The converse is easily seen to be false. I am interested in understanding how often the converse fails. From my view, one way to make this precise is as follows. Let $H_n$ denote the set of Hamiltonian graphs on $n$-vertices. What can we say about the probability $$p_n = P(\deg(v) \geq \frac{n}{2},\forall v\in G \mid G \in H_n)$$

I am mainly interested in what happens as $n \to \infty$. For example, I think it might be interesting if Dirac's theorem becomes necessary and sufficient if we take $n$ large enough. One could also investigate analogous question for other theorems that give sufficient conditions for $G$ to be Hamiltonian (Ore's theorem, Posa's theorem). However, Dirac's seemed the simplest to investigate. 

Is there any literature on questions resembling this?

Thanks.

**Edit:**

The previous calculations for various $p_n$ values were clearly wrong and have been removed.